Why does the Moon appear to change shape?

The Moon changes its appearance because we see different portions of its sunlit half. Its shape remains round. The dark-looking part of the disk has not broken away; it is the part of the visible surface that sunlight is not directly lighting.

The upper picture looks down on the Earth–Moon arrangement. Sunlight arrives from the left. At each illustrated position, the half of the Moon facing left receives sunlight. What changes is the direction from which someone on Earth sees that illuminated surface.

Near new Moon, the lit half mostly faces away from Earth. At a quarter phase, half of the visible disk is lit. Near full Moon, the lit half mostly faces Earth. The lower disks show these three appearances. Their orientation is a simplified viewing convention; the bright side can look tilted in the real sky.

An ordinary phase is not a piece of Earth’s shadow crossing the Moon. A lunar eclipse occurs when the Moon actually passes through that shadow. The Moon’s tilted orbit usually carries it above or below the alignment needed for an eclipse.

Keeping two views separate resolves the puzzle: about half of the whole Moon receives sunlight, while the fraction of its lit surface that faces us changes. The phase is a relationship between illumination and viewpoint.